p -derivations (Joyal, Buium) A p -derivation on R is a function δ : R → R modeled on δ ( x ) = x ′ = ψ ( x ) − x p , p i.e., satisfying all the axioms it does when ψ is a Frobenius lift and R is p -torsion free: p − 1 1 � p � � x i y p − i δ ( x + y ) = δ ( x ) + δ ( y ) − p i i = 1 δ ( xy ) = δ ( x ) y p + x p δ ( y ) + p δ ( x ) δ ( y ) δ ( 0 ) = 0 δ ( 1 ) = 0 Leibniz rules for multiplication and addition: δ ( x ) = x ′ = “ ∂ x /∂ p ” Category: δ -rings { p -derivations on R } ∼ → { Frobenius lifts on R } , if R is p -tor-free 4
Divided power series = cofree differential ring Consider usual derivations d , instead of p -derivations δ : U � Ring { d -rings } 5
� � Divided power series = cofree differential ring Consider usual derivations d , instead of p -derivations δ : ⊥ � Ring { d -rings } ⊥ W diff 5
� � Divided power series = cofree differential ring Consider usual derivations d , instead of p -derivations δ : ⊥ � Ring { d -rings } ⊥ W diff t n � � W diff ( R ) = � a n n ! | a n ∈ R , d = d / dt n 5
� � Divided power series = cofree differential ring Consider usual derivations d , instead of p -derivations δ : ⊥ � Ring { d -rings } ⊥ W diff t n � � W diff ( R ) = � a n n ! | a n ∈ R , d = d / dt n = { ( a 0 , a 1 , . . . ) } , d = shift Multiplication law at the n -th component is given by the Leibniz rule for d ◦ n ( xy ) : ( a 0 , . . . ) × ( b 0 , . . . ) = ( a 0 b 0 , a 0 b 1 + a 1 b 0 , a 0 b 2 + 2 a 1 b 1 + a 2 b 0 , . . . ) 5
Witt vectors = cofree δ -ring (Joyal) � Ring { δ -rings } 6
� � Witt vectors = cofree δ -ring (Joyal) ⊥ � Ring { δ -rings } ⊥ W 6
� � Witt vectors = cofree δ -ring (Joyal) ⊥ � Ring { δ -rings } ⊥ W W ( R ) = R × R × R × · · · , δ ( a 0 , a 1 , . . . ) = ( a 1 , a 2 , . . . ) Mulitiplication at the n -th component is again given by the Leibniz rule for δ ◦ n ( xy ) , but now the same is true for addition! 6
� � Witt vectors = cofree δ -ring (Joyal) ⊥ � Ring { δ -rings } ⊥ W W ( R ) = R × R × R × · · · , δ ( a 0 , a 1 , . . . ) = ( a 1 , a 2 , . . . ) Mulitiplication at the n -th component is again given by the Leibniz rule for δ ◦ n ( xy ) , but now the same is true for addition! 1 � p � � 0 b p − i a i ( a 0 , a 1 , . . . ) + ( b 0 , b 1 , . . . ) = ( a 0 + b 0 , a 1 + b 1 − , . . . ) 0 p i i ( a 0 , a 1 , . . . ) × ( b 0 , b 1 , . . . ) = ( a 0 b 0 , a 1 b p 0 + a p 0 b 1 + pa 1 b 1 , . . . ) 6
� � Witt vectors = cofree δ -ring (Joyal) ⊥ � Ring { δ -rings } ⊥ W W ( R ) = R × R × R × · · · , δ ( a 0 , a 1 , . . . ) = ( a 1 , a 2 , . . . ) Mulitiplication at the n -th component is again given by the Leibniz rule for δ ◦ n ( xy ) , but now the same is true for addition! 1 � p � � 0 b p − i a i ( a 0 , a 1 , . . . ) + ( b 0 , b 1 , . . . ) = ( a 0 + b 0 , a 1 + b 1 − , . . . ) 0 p i i ( a 0 , a 1 , . . . ) × ( b 0 , b 1 , . . . ) = ( a 0 b 0 , a 1 b p 0 + a p 0 b 1 + pa 1 b 1 , . . . ) Leibniz rules: p − 1 1 � p � � x i y p − i δ ( x + y ) = δ ( x ) + δ ( y ) − p i i = 1 δ ( xy ) = δ ( x ) y p + x p δ ( y ) + p δ ( x ) δ ( y ) 6
Remarks ◮ Warning: The ring structure on R × R × · · · above is not equal to the Witt vector ring structure as it is usually defined! Only uniquely isomorphic to it. 7
Remarks ◮ Warning: The ring structure on R × R × · · · above is not equal to the Witt vector ring structure as it is usually defined! Only uniquely isomorphic to it. ◮ Ex: W ( Z / p Z ) ∼ = ring Z p of p -adic integers 7
Remarks ◮ Warning: The ring structure on R × R × · · · above is not equal to the Witt vector ring structure as it is usually defined! Only uniquely isomorphic to it. ◮ Ex: W ( Z / p Z ) ∼ = ring Z p of p -adic integers ← characteristic 0! 7
Remarks ◮ Warning: The ring structure on R × R × · · · above is not equal to the Witt vector ring structure as it is usually defined! Only uniquely isomorphic to it. ◮ Ex: W ( Z / p Z ) ∼ = ring Z p of p -adic integers ← characteristic 0! ◮ More generally, the map Z → W ( R ) is injective unless R = 0. 7
Remarks ◮ Warning: The ring structure on R × R × · · · above is not equal to the Witt vector ring structure as it is usually defined! Only uniquely isomorphic to it. ◮ Ex: W ( Z / p Z ) ∼ = ring Z p of p -adic integers ← characteristic 0! ◮ More generally, the map Z → W ( R ) is injective unless R = 0. ◮ Witt vectors are a machine for functorially lifting rings from characteristic p to characteristic 0 7
Remarks ◮ Warning: The ring structure on R × R × · · · above is not equal to the Witt vector ring structure as it is usually defined! Only uniquely isomorphic to it. ◮ Ex: W ( Z / p Z ) ∼ = ring Z p of p -adic integers ← characteristic 0! ◮ More generally, the map Z → W ( R ) is injective unless R = 0. ◮ Witt vectors are a machine for functorially lifting rings from characteristic p to characteristic 0 ◮ Better: Witt vectors are a machine for adding a Frobenius lift to your ring, interpreted in an intelligent way 7
de Rham–Witt complex (Bloch, Deligne, Illusie, 1970s–) ◮ de Rham cohomology has problems in characteristic p : any function f p is a closed 0-form d ( f p ) = pf p − 1 df = 0 8
de Rham–Witt complex (Bloch, Deligne, Illusie, 1970s–) ◮ de Rham cohomology has problems in characteristic p : any function f p is a closed 0-form d ( f p ) = pf p − 1 df = 0 ◮ One can lift rings/varieties to characteristic 0 using Witt vectors 8
de Rham–Witt complex (Bloch, Deligne, Illusie, 1970s–) ◮ de Rham cohomology has problems in characteristic p : any function f p is a closed 0-form d ( f p ) = pf p − 1 df = 0 ◮ One can lift rings/varieties to characteristic 0 using Witt vectors ◮ . . . the de Rham–Witt complex W Ω ∗ X 8
de Rham–Witt complex (Bloch, Deligne, Illusie, 1970s–) ◮ de Rham cohomology has problems in characteristic p : any function f p is a closed 0-form d ( f p ) = pf p − 1 df = 0 ◮ One can lift rings/varieties to characteristic 0 using Witt vectors ◮ . . . the de Rham–Witt complex W Ω ∗ X ◮ Calculates crystalline cohomology (with its Frobenius operator) 8
de Rham–Witt complex (Bloch, Deligne, Illusie, 1970s–) ◮ de Rham cohomology has problems in characteristic p : any function f p is a closed 0-form d ( f p ) = pf p − 1 df = 0 ◮ One can lift rings/varieties to characteristic 0 using Witt vectors ◮ . . . the de Rham–Witt complex W Ω ∗ X ◮ Calculates crystalline cohomology (with its Frobenius operator) ◮ Thus, if one is sufficiently enlightened, the concept of Frobenius lift, or p -derivation, leads automatically to crystalline cohomology. 8
II. Generalized symmetries ( Tall–Wraith, Bergman–Hausknecht, Wieland & me, Stacey–Whitehouse ) C = a category of ‘algebras’ (rings, groups, Lie algebras,. . . ) 9
II. Generalized symmetries ( Tall–Wraith, Bergman–Hausknecht, Wieland & me, Stacey–Whitehouse ) C = a category of ‘algebras’ (rings, groups, Lie algebras,. . . ) U : D → C comonadic, where the comonad W is representable: Hom C ( P , R ) = underlying set of W ( R ) 9
II. Generalized symmetries ( Tall–Wraith, Bergman–Hausknecht, Wieland & me, Stacey–Whitehouse ) C = a category of ‘algebras’ (rings, groups, Lie algebras,. . . ) U : D → C comonadic, where the comonad W is representable: Hom C ( P , R ) = underlying set of W ( R ) P = U ( free object of D on one generator ) = { natural 1-ary operations on objects of D } 9
II. Generalized symmetries ( Tall–Wraith, Bergman–Hausknecht, Wieland & me, Stacey–Whitehouse ) C = a category of ‘algebras’ (rings, groups, Lie algebras,. . . ) U : D → C comonadic, where the comonad W is representable: Hom C ( P , R ) = underlying set of W ( R ) P = U ( free object of D on one generator ) = { natural 1-ary operations on objects of D } ◮ G -rings → Ring , G = group or monoid P = { polynomials in elements of G } = Sym ( Z G ) 9
II. Generalized symmetries ( Tall–Wraith, Bergman–Hausknecht, Wieland & me, Stacey–Whitehouse ) C = a category of ‘algebras’ (rings, groups, Lie algebras,. . . ) U : D → C comonadic, where the comonad W is representable: Hom C ( P , R ) = underlying set of W ( R ) P = U ( free object of D on one generator ) = { natural 1-ary operations on objects of D } ◮ G -rings → Ring , G = group or monoid P = { polynomials in elements of G } = Sym ( Z G ) ◮ d -rings → Ring , W = W diff = divided power series functor P = Z [ e , d , d ◦ 2 , . . . ] = differential operators 9
II. Generalized symmetries ( Tall–Wraith, Bergman–Hausknecht, Wieland & me, Stacey–Whitehouse ) C = a category of ‘algebras’ (rings, groups, Lie algebras,. . . ) U : D → C comonadic, where the comonad W is representable: Hom C ( P , R ) = underlying set of W ( R ) P = U ( free object of D on one generator ) = { natural 1-ary operations on objects of D } ◮ G -rings → Ring , G = group or monoid P = { polynomials in elements of G } = Sym ( Z G ) ◮ d -rings → Ring , W = W diff = divided power series functor P = Z [ e , d , d ◦ 2 , . . . ] = differential operators ◮ δ -rings → Ring , W = Witt vector functor P = Z [ e , δ, δ ◦ 2 , . . . ] = ‘ p -differential operators’ 9
II. Generalized symmetries ( Tall–Wraith, Bergman–Hausknecht, Wieland & me, Stacey–Whitehouse ) C = a category of ‘algebras’ (rings, groups, Lie algebras,. . . ) U : D → C comonadic, where the comonad W is representable: Hom C ( P , R ) = underlying set of W ( R ) P = U ( free object of D on one generator ) = { natural 1-ary operations on objects of D } ◮ G -rings → Ring , G = group or monoid P = { polynomials in elements of G } = Sym ( Z G ) ◮ d -rings → Ring , W = W diff = divided power series functor P = Z [ e , d , d ◦ 2 , . . . ] = differential operators ◮ δ -rings → Ring , W = Witt vector functor P = Z [ e , δ, δ ◦ 2 , . . . ] = ‘ p -differential operators’ A composition object of C is an object P of C plus a comonad structure on the functor it represents. (‘Tall–Wraith monad object’) 9
Generalized symmetries, continued ◮ Since P is the set of natural operations on objects of D , 10
Generalized symmetries, continued ◮ Since P is the set of natural operations on objects of D , we may think of it as a system of generalized symmetries which may act on objects of C 10
Generalized symmetries, continued ◮ Since P is the set of natural operations on objects of D , we may think of it as a system of generalized symmetries which may act on objects of C ◮ It is closed under composition and the all the operations of C ◮ E.g.: differential operators Z [ e , d , d ◦ 2 , . . . ] 10
Generalized symmetries, continued ◮ Since P is the set of natural operations on objects of D , we may think of it as a system of generalized symmetries which may act on objects of C ◮ It is closed under composition and the all the operations of C ◮ E.g.: differential operators Z [ e , d , d ◦ 2 , . . . ] ◮ An element f in a composition ring P is linear if it acts additively whenever P acts on a ring 10
Generalized symmetries, continued ◮ Since P is the set of natural operations on objects of D , we may think of it as a system of generalized symmetries which may act on objects of C ◮ It is closed under composition and the all the operations of C ◮ E.g.: differential operators Z [ e , d , d ◦ 2 , . . . ] ◮ An element f in a composition ring P is linear if it acts additively whenever P acts on a ring ◮ The p -derivation δ ∈ Z [ e , δ, δ ◦ 2 , . . . ] is not linear, but the Frobenius lift ψ = e p + p δ is. 10
Generalized symmetries, continued ◮ Since P is the set of natural operations on objects of D , we may think of it as a system of generalized symmetries which may act on objects of C ◮ It is closed under composition and the all the operations of C ◮ E.g.: differential operators Z [ e , d , d ◦ 2 , . . . ] ◮ An element f in a composition ring P is linear if it acts additively whenever P acts on a ring ◮ The p -derivation δ ∈ Z [ e , δ, δ ◦ 2 , . . . ] is not linear, but the Frobenius lift ψ = e p + p δ is. ◮ In fact, the composition ring Z [ e , δ, δ ◦ 2 , . . . ] cannot be generated by linear operators! It is fundamentally nonlinear. 10
Imperative task #1 Given C, determine all its composition objets P 11
Imperative task #1 Given C, determine all its composition objets P ◮ R -modules: P = (noncomm.) ring with a map R → P 11
Imperative task #1 Given C, determine all its composition objets P ◮ R -modules: P = (noncomm.) ring with a map R → P ◮ Groups (Kan): P is the free group on some monoid M . So generalized symmetries are words in endomorphisms 11
Imperative task #1 Given C, determine all its composition objets P ◮ R -modules: P = (noncomm.) ring with a map R → P ◮ Groups (Kan): P is the free group on some monoid M . So generalized symmetries are words in endomorphisms ◮ Monoids (Bergman–Hausknecht): Generalized symmetries are words in endomorphisms and anti-endomorphisms (but there can be relations!) 11
Imperative task #1 Given C, determine all its composition objets P ◮ R -modules: P = (noncomm.) ring with a map R → P ◮ Groups (Kan): P is the free group on some monoid M . So generalized symmetries are words in endomorphisms ◮ Monoids (Bergman–Hausknecht): Generalized symmetries are words in endomorphisms and anti-endomorphisms (but there can be relations!) ◮ Magnus Carlson (2016): If K is a field of characteristic 0, all composition objects of CAlg K are freely generated by bialgebras of linear operators! 11
Imperative task #1 Given C, determine all its composition objets P ◮ R -modules: P = (noncomm.) ring with a map R → P ◮ Groups (Kan): P is the free group on some monoid M . So generalized symmetries are words in endomorphisms ◮ Monoids (Bergman–Hausknecht): Generalized symmetries are words in endomorphisms and anti-endomorphisms (but there can be relations!) ◮ Magnus Carlson (2016): If K is a field of characteristic 0, all composition objects of CAlg K are freely generated by bialgebras of linear operators! ◮ Is it possible to classify all composition objects in Ring ? 11
Imperative task #1 Given C, determine all its composition objets P ◮ R -modules: P = (noncomm.) ring with a map R → P ◮ Groups (Kan): P is the free group on some monoid M . So generalized symmetries are words in endomorphisms ◮ Monoids (Bergman–Hausknecht): Generalized symmetries are words in endomorphisms and anti-endomorphisms (but there can be relations!) ◮ Magnus Carlson (2016): If K is a field of characteristic 0, all composition objects of CAlg K are freely generated by bialgebras of linear operators! ◮ Is it possible to classify all composition objects in Ring ? ◮ Carlson: Yes, if we allow denominators ◮ Buium: Some positive classification results for composition rings generated by a single operator ◮ All known examples come from linear operators or lifting Frobenius-like constructions from char p to char 0. 11
Imperative task #2 (with Garner) Given C and an object X of interest. ◮ Everyone: To understand X , it is important to know all of its symmetries 12
Imperative task #2 (with Garner) Given C and an object X of interest. ◮ Everyone: To understand X , it is important to know all of its symmetries ◮ Also everyone: If X is a manifold/scheme/ring/. . . , this should be understood to include infinitesimal symmetries (vector fields and derivations) 12
Imperative task #2 (with Garner) Given C and an object X of interest. ◮ Everyone: To understand X , it is important to know all of its symmetries ◮ Also everyone: If X is a manifold/scheme/ring/. . . , this should be understood to include infinitesimal symmetries (vector fields and derivations) ◮ But it should really include all generalized symmetries! 12
Imperative task #2 (with Garner) Given C and an object X of interest. ◮ Everyone: To understand X , it is important to know all of its symmetries ◮ Also everyone: If X is a manifold/scheme/ring/. . . , this should be understood to include infinitesimal symmetries (vector fields and derivations) ◮ But it should really include all generalized symmetries! ◮ Thm (Bird): Given an object X of C, there is a terminal composition object acting on X . 12
Imperative task #2 (with Garner) Given C and an object X of interest. ◮ Everyone: To understand X , it is important to know all of its symmetries ◮ Also everyone: If X is a manifold/scheme/ring/. . . , this should be understood to include infinitesimal symmetries (vector fields and derivations) ◮ But it should really include all generalized symmetries! ◮ Thm (Bird): Given an object X of C, there is a terminal composition object acting on X . ◮ Call it END ( X ) , the full symmetry composition object of X . 12
Imperative task #2 (with Garner) Given C and an object X of interest. ◮ Everyone: To understand X , it is important to know all of its symmetries ◮ Also everyone: If X is a manifold/scheme/ring/. . . , this should be understood to include infinitesimal symmetries (vector fields and derivations) ◮ But it should really include all generalized symmetries! ◮ Thm (Bird): Given an object X of C, there is a terminal composition object acting on X . ◮ Call it END ( X ) , the full symmetry composition object of X . If you are interested in X , you must determine END ( X ) , and then you should try to work “ END ( X ) -equivariantly” 12
Imperative task #2 (with Garner) Given C and an object X of interest. ◮ Everyone: To understand X , it is important to know all of its symmetries ◮ Also everyone: If X is a manifold/scheme/ring/. . . , this should be understood to include infinitesimal symmetries (vector fields and derivations) ◮ But it should really include all generalized symmetries! ◮ Thm (Bird): Given an object X of C, there is a terminal composition object acting on X . ◮ Call it END ( X ) , the full symmetry composition object of X . If you are interested in X , you must determine END ( X ) , and then you should try to work “ END ( X ) -equivariantly” ◮ END ( Z ) ? = { quasi-polynomials Z → Z } (with Garner) 12
Imperative task #2 (with Garner) Given C and an object X of interest. ◮ Everyone: To understand X , it is important to know all of its symmetries ◮ Also everyone: If X is a manifold/scheme/ring/. . . , this should be understood to include infinitesimal symmetries (vector fields and derivations) ◮ But it should really include all generalized symmetries! ◮ Thm (Bird): Given an object X of C, there is a terminal composition object acting on X . ◮ Call it END ( X ) , the full symmetry composition object of X . If you are interested in X , you must determine END ( X ) , and then you should try to work “ END ( X ) -equivariantly” ◮ END ( Z ) ? = { quasi-polynomials Z → Z } (with Garner) ◮ END ( F p [ t ]) = ?. Includes derivation d / dt , t -derivation f �→ ( f − f q ) / t ,. . . 12
III. Generalized-equivariant algebriac geometry Principal categories of algebraic geometry: Ring op = Aff ⊂ Sch ⊂ AlgSp ⊂ Sh ét ( Aff ) ⊂ PSh ( Aff ) 13
III. Generalized-equivariant algebriac geometry Principal categories of algebraic geometry: Ring op = Aff ⊂ Sch ⊂ AlgSp ⊂ Sh ét ( Aff ) ⊂ PSh ( Aff ) Is it possible to extend the theory of generalized symmetries from Ring to non-affine schemes? 13
III. Generalized-equivariant algebriac geometry Principal categories of algebraic geometry: Ring op = Aff ⊂ Sch ⊂ AlgSp ⊂ Sh ét ( Aff ) ⊂ PSh ( Aff ) Is it possible to extend the theory of generalized symmetries from Ring to non-affine schemes? ◮ Monoid and Lie algebra actions (linear symmetries) are OK: G -schemes, g -schemes ◮ Can this be done for p -derivations and similar non-linear symmetries? (Yes! See below.) 13
III. Generalized-equivariant algebriac geometry Principal categories of algebraic geometry: Ring op = Aff ⊂ Sch ⊂ AlgSp ⊂ Sh ét ( Aff ) ⊂ PSh ( Aff ) Is it possible to extend the theory of generalized symmetries from Ring to non-affine schemes? ◮ Monoid and Lie algebra actions (linear symmetries) are OK: G -schemes, g -schemes ◮ Can this be done for p -derivations and similar non-linear symmetries? (Yes! See below.) ◮ Can this be done for every composition ring? 13
III. Generalized-equivariant algebriac geometry Principal categories of algebraic geometry: Ring op = Aff ⊂ Sch ⊂ AlgSp ⊂ Sh ét ( Aff ) ⊂ PSh ( Aff ) Is it possible to extend the theory of generalized symmetries from Ring to non-affine schemes? ◮ Monoid and Lie algebra actions (linear symmetries) are OK: G -schemes, g -schemes ◮ Can this be done for p -derivations and similar non-linear symmetries? (Yes! See below.) ◮ Can this be done for every composition ring? ◮ Could there some kind of new generalized symmetry structures that exist only at the non-affine level? 13
δ -structures on schemes (Greenberg, Buium, me) Given a functor X : Ring → Set , define W n ∗ ( X ): C �→ X ( W n ( C )) , where W n ( C ) is the ring of truncated Witt vectors ( a 0 , . . . , a n ) . 14
δ -structures on schemes (Greenberg, Buium, me) Given a functor X : Ring → Set , define W n ∗ ( X ): C �→ X ( W n ( C )) , where W n ( C ) is the ring of truncated Witt vectors ( a 0 , . . . , a n ) . ◮ W n ( C ) is analogous to the truncated power series ring. So W n ∗ ( X ) is a Witt vector analogue of the n -th jet space, the “arithmetic jet space” 14
δ -structures on schemes (Greenberg, Buium, me) Given a functor X : Ring → Set , define W n ∗ ( X ): C �→ X ( W n ( C )) , where W n ( C ) is the ring of truncated Witt vectors ( a 0 , . . . , a n ) . ◮ W n ( C ) is analogous to the truncated power series ring. So W n ∗ ( X ) is a Witt vector analogue of the n -th jet space, the “arithmetic jet space” Thm: If X is a scheme, then so is W n ∗ ( X ) . Likewise for algebraic spaces and sheaves in the étale topology. 14
δ -structures on schemes (Greenberg, Buium, me) Given a functor X : Ring → Set , define W n ∗ ( X ): C �→ X ( W n ( C )) , where W n ( C ) is the ring of truncated Witt vectors ( a 0 , . . . , a n ) . ◮ W n ( C ) is analogous to the truncated power series ring. So W n ∗ ( X ) is a Witt vector analogue of the n -th jet space, the “arithmetic jet space” Thm: If X is a scheme, then so is W n ∗ ( X ) . Likewise for algebraic spaces and sheaves in the étale topology. ◮ This allows us to extend the theory of p -derivations, δ -structures, and Witt vectors from rings to schemes → “ δ -equivariant algebraic geometry” 14
δ -structures on schemes (Greenberg, Buium, me) Given a functor X : Ring → Set , define W n ∗ ( X ): C �→ X ( W n ( C )) , where W n ( C ) is the ring of truncated Witt vectors ( a 0 , . . . , a n ) . ◮ W n ( C ) is analogous to the truncated power series ring. So W n ∗ ( X ) is a Witt vector analogue of the n -th jet space, the “arithmetic jet space” Thm: If X is a scheme, then so is W n ∗ ( X ) . Likewise for algebraic spaces and sheaves in the étale topology. ◮ This allows us to extend the theory of p -derivations, δ -structures, and Witt vectors from rings to schemes → “ δ -equivariant algebraic geometry” ◮ The proof (Illusie, van der Kallen, Langer–Zink, me) is not formal! 14
Hilbert’s 12th Problem Given a finite extension K / Q , is there an explicit description of K ab , its maximal Galois extension with abelian Galois group? ◮ K = Q : Yes, the Kronecker–Weber theorem (1853–1896): adjoin all roots of unity exp ( 2 π i n ) to Q 15
Hilbert’s 12th Problem Given a finite extension K / Q , is there an explicit description of K ab , its maximal Galois extension with abelian Galois group? ◮ K = Q : Yes, the Kronecker–Weber theorem (1853–1896): adjoin all roots of unity exp ( 2 π i n ) to Q √ ◮ K = Q ( − d ) , d > 0: Yes, Kronecker’s Jugendtraum (1850s–1920): adjoin certain special values of elliptic and √ modular functions to Q ( − d ) 15
Hilbert’s 12th Problem Given a finite extension K / Q , is there an explicit description of K ab , its maximal Galois extension with abelian Galois group? ◮ K = Q : Yes, the Kronecker–Weber theorem (1853–1896): adjoin all roots of unity exp ( 2 π i n ) to Q √ ◮ K = Q ( − d ) , d > 0: Yes, Kronecker’s Jugendtraum (1850s–1920): adjoin certain special values of elliptic and √ modular functions to Q ( − d ) ◮ Nowadays, people usually express them in terms of adjoining the coordinates of torsion points on commutative group schemes, instead of special values of transcendental functions 15
Hilbert’s 12th Problem Given a finite extension K / Q , is there an explicit description of K ab , its maximal Galois extension with abelian Galois group? ◮ K = Q : Yes, the Kronecker–Weber theorem (1853–1896): adjoin all roots of unity exp ( 2 π i n ) to Q √ ◮ K = Q ( − d ) , d > 0: Yes, Kronecker’s Jugendtraum (1850s–1920): adjoin certain special values of elliptic and √ modular functions to Q ( − d ) ◮ Nowadays, people usually express them in terms of adjoining the coordinates of torsion points on commutative group schemes, instead of special values of transcendental functions ◮ No other answers to H12 are known. But H12 is imprecise! 15
Hilbert’s 12th Problem Given a finite extension K / Q , is there an explicit description of K ab , its maximal Galois extension with abelian Galois group? ◮ K = Q : Yes, the Kronecker–Weber theorem (1853–1896): adjoin all roots of unity exp ( 2 π i n ) to Q √ ◮ K = Q ( − d ) , d > 0: Yes, Kronecker’s Jugendtraum (1850s–1920): adjoin certain special values of elliptic and √ modular functions to Q ( − d ) ◮ Nowadays, people usually express them in terms of adjoining the coordinates of torsion points on commutative group schemes, instead of special values of transcendental functions ◮ No other answers to H12 are known. But H12 is imprecise! ◮ Class field theory (Hilbert–Takagi–Artin, 1896–1927) gives an explicit description of Gal ( K ab / K ) —but not of K ab ! 15
Hilbert’s 12th Problem Given a finite extension K / Q , is there an explicit description of K ab , its maximal Galois extension with abelian Galois group? ◮ K = Q : Yes, the Kronecker–Weber theorem (1853–1896): adjoin all roots of unity exp ( 2 π i n ) to Q √ ◮ K = Q ( − d ) , d > 0: Yes, Kronecker’s Jugendtraum (1850s–1920): adjoin certain special values of elliptic and √ modular functions to Q ( − d ) ◮ Nowadays, people usually express them in terms of adjoining the coordinates of torsion points on commutative group schemes, instead of special values of transcendental functions ◮ No other answers to H12 are known. But H12 is imprecise! ◮ Class field theory (Hilbert–Takagi–Artin, 1896–1927) gives an explicit description of Gal ( K ab / K ) —but not of K ab ! ◮ New idea: Use periodic points on Λ K -schemes instead! 15
Λ K -structures Fix a finite extension K / Q . Let O K denote its subring of algebraic integers. Let R be an O K -algebra. ◮ A Λ K -structure on R is a commuting family of endomorphisms ψ p , one for each nonzero prime ideal p ⊂ O K such that ψ p ( x ) ≡ x N ( p ) mod p R , where N ( p ) = |O K / p | . 16
Λ K -structures Fix a finite extension K / Q . Let O K denote its subring of algebraic integers. Let R be an O K -algebra. ◮ A Λ K -structure on R is a commuting family of endomorphisms ψ p , one for each nonzero prime ideal p ⊂ O K such that ψ p ( x ) ≡ x N ( p ) mod p R , where N ( p ) = |O K / p | . ◮ Similarly for schemes. 16
Λ K -structures Fix a finite extension K / Q . Let O K denote its subring of algebraic integers. Let R be an O K -algebra. ◮ A Λ K -structure on R is a commuting family of endomorphisms ψ p , one for each nonzero prime ideal p ⊂ O K such that ψ p ( x ) ≡ x N ( p ) mod p R , where N ( p ) = |O K / p | . ◮ Similarly for schemes. ◮ If there is nontrivial torsion, we have to interpret all this in the enlightened way, as with Frobenius lifts at a single prime. 16
Λ K -structures Fix a finite extension K / Q . Let O K denote its subring of algebraic integers. Let R be an O K -algebra. ◮ A Λ K -structure on R is a commuting family of endomorphisms ψ p , one for each nonzero prime ideal p ⊂ O K such that ψ p ( x ) ≡ x N ( p ) mod p R , where N ( p ) = |O K / p | . ◮ Similarly for schemes. ◮ If there is nontrivial torsion, we have to interpret all this in the enlightened way, as with Frobenius lifts at a single prime. ◮ → composition O K -algebra Λ K , again nonlinear! 16
Λ K -structures Fix a finite extension K / Q . Let O K denote its subring of algebraic integers. Let R be an O K -algebra. ◮ A Λ K -structure on R is a commuting family of endomorphisms ψ p , one for each nonzero prime ideal p ⊂ O K such that ψ p ( x ) ≡ x N ( p ) mod p R , where N ( p ) = |O K / p | . ◮ Similarly for schemes. ◮ If there is nontrivial torsion, we have to interpret all this in the enlightened way, as with Frobenius lifts at a single prime. ◮ → composition O K -algebra Λ K , again nonlinear! ◮ Wilkerson, Joyal: Λ Q -ring = λ -ring as in K-theory 16
Λ K -structures and Hilbert’s 12th Problem (with de Smit) ◮ Given a Λ K -scheme X , a point x is periodic if ψ p ( x ) is periodic as a function of p (in a certain technical sense) 17
Λ K -structures and Hilbert’s 12th Problem (with de Smit) ◮ Given a Λ K -scheme X , a point x is periodic if ψ p ( x ) is periodic as a function of p (in a certain technical sense) 17
Λ K -structures and Hilbert’s 12th Problem (with de Smit) ◮ Given a Λ K -scheme X , a point x is periodic if ψ p ( x ) is periodic as a function of p (in a certain technical sense) ◮ E.g. K = Q , X ( C ) = C ∗ , ψ p ( x ) = x p Then x is periodic ⇔ x is a root of unity 17
Λ K -structures and Hilbert’s 12th Problem (with de Smit) ◮ Given a Λ K -scheme X , a point x is periodic if ψ p ( x ) is periodic as a function of p (in a certain technical sense) ◮ E.g. K = Q , X ( C ) = C ∗ , ψ p ( x ) = x p Then x is periodic ⇔ x is a root of unity ◮ Thm: The coordinates of the periodic points of X generate an abelian extension of K (if X is of finite type). 17
Λ K -structures and Hilbert’s 12th Problem (with de Smit) ◮ Given a Λ K -scheme X , a point x is periodic if ψ p ( x ) is periodic as a function of p (in a certain technical sense) ◮ E.g. K = Q , X ( C ) = C ∗ , ψ p ( x ) = x p Then x is periodic ⇔ x is a root of unity ◮ Thm: The coordinates of the periodic points of X generate an abelian extension of K (if X is of finite type). ◮ An extension L / K is Λ -geometric if it can be generated by the periodic points of some such X 17
Λ K -structures and Hilbert’s 12th Problem (with de Smit) ◮ Given a Λ K -scheme X , a point x is periodic if ψ p ( x ) is periodic as a function of p (in a certain technical sense) ◮ E.g. K = Q , X ( C ) = C ∗ , ψ p ( x ) = x p Then x is periodic ⇔ x is a root of unity ◮ Thm: The coordinates of the periodic points of X generate an abelian extension of K (if X is of finite type). ◮ An extension L / K is Λ -geometric if it can be generated by the periodic points of some such X ◮ This allows for a yes/no formulation of Hilbert’s 12th Problem: Is K ab / K a Λ -geometric extension? 17
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